Final answer:
To prove that 0.72 recurring equals 8/11, define x as 0.72 recurring, multiply by 100 to obtain an equation, subtract the original number from this new equation, solve for x and simplify to demonstrate that x equals 8/11.
Step-by-step explanation:
To prove algebraically that 0.72 recurring is equal to 8/11, we'll use variables and algebraic manipulation. Let's define x to be the repeating decimal 0.72 (where 72 is the repeating part).
First, we write it as:
x = 0.727272...
To eliminate the recurring part, we can multiply x by 100 since the repeating part is two digits long:
100x = 72.727272...
Now, we subtract the original x from this new equation:
100x - x = 72.727272... - 0.727272...
Which simplifies to:
99x = 72
Dividing both sides by 99 to solve for x:
x = 72 / 99
Both numerator and denominator are divisible by 9, so we simplify:
x = 8 / 11
Thus, we've algebraically proven that the recurring decimal 0.72 is indeed equal to 8/11.